In recent years, the theory of variable exponent function spaces has attracted much more attention (see [1-8] for example), since Kováčik and Pákosník [1] introduced the variable exponents Lebesgue and Sobolev spaces, which are the fundamental works of variable exponent function spaces. The function spaces with variable exponent have been applied widely in the image processing, fluid mechanics and partial differential equations with non-standard growth(see [9-13] for example).

Let $\Omega\in L^s(\mathbb{S}^{n-1})$ be homogeneous of degree zero on $\mathbb{R}^{n}, $ where $\mathbb{S}^{n-1}$ denotes the unit sphere of $\mathbb{R}^{n}(n\geq 2)$ equipped with the normalized Lebesgue measure $d\sigma$ and $s\geq 1.$ For $0<α<n, $ then the fractional integral with rough kernel is defined by

$ T_{\Omega, \alpha}(f)(x)=\int_{\mathbb{R}^{n}}\frac{\Omega(x-y)}{|x-y|^{n-\alpha}}f(y)dy. $

(1.1)

Moreover, let $b\in L_{loc}^1(\mathbb{R}^{n}).$ Then the commutator generated by $T_{\Omega, \alpha}$ and $b$ can be defined as follows

$ [b, T_{\Omega, \alpha}](f)(x)=\int_{\mathbb{R}^{n}}[b(x)-b(y)]\frac{\Omega(x-y)}{|x-y|^{n-\alpha}}f(y)dy. $

(1.2)

Recently, Tan and Liu [14] studied the boundedness of $T_{\Omega, \alpha}$ on the variable exponent Lebesgue, Hardy and Herz-type Hardy spaces. Wang [15] et al. obtained the boundedness of $T_{\Omega, \alpha}$ and its commutator $[b, T_{\Omega, \alpha}]$ on Morrey-Herz space with variable exponent. Tan and Zhao [16] established the boundedness of $T_{\Omega, \alpha}$ and its commutator $[b, T_{\Omega, \alpha}]$ in variable exponent Morrey spaces. In 2016, Long and Han [17] established the boundedness of the maximal operator, potential operator and singular operator of Calderón-Zygmund type in the vanishing generalized Morrey spaces with variable exponent.

Inspired by [14-17], in the paper, we will consider the boundedness of $T_{\Omega, \alpha}$ and its commutator $[b, T_{\Omega, \alpha}]$ generated by $T_{\Omega, \alpha}$ and BMO functions in vanishing generalized Morrey spaces with variable exponent on unbounded sets.

Notation   Throughout this paper, $\mathbb{R}^{n}$ is the $n-$dimensional Euclidean space, $\chi_{A}$ is the characteristic function of a set $A\subseteq\mathbb{R}^{n}.$ $C$ is the positive constant, which may have different values even in the same line. $A \lesssim B$ means that $A \leq CB$ with some positive constant $C$ independent of appropriate quantities and if $A \lesssim B$ and $B\lesssim A, $ we write $A \thickapprox B.$

Now, let us recall some necessary definitions and notations.

Definition 1.1   Let $D\subset\mathbb{R}^{n}$ be an open set and $p(\cdot): D\rightarrow[1, \infty) $ be a measurable function. Then, $\;L^{p(\cdot)}(D)$ denotes the set of all measurable functions $f$ on $D$ such that for some $\lambda>0$,

$ \int_{D}\bigg(\frac{|f(x)|}{\lambda}\bigg)^{p(x)}dx<\infty. $

This set becomes a Banach function space when equipped with the Luxemburg-Nakano norm

$ \|f\|_{L^{p(\cdot)}(D)}=\inf\bigg\{\lambda>0:\int_{D}\bigg(\frac{|f(x)|}{\lambda}\bigg)^{p(x)}dx\leq 1\bigg\}. $

$L^{p(\cdot)}(D)$ is regarded as variable exponent Lebesgue space. And, if $p(x)=p$ is a positive constant, then $L^{p(\cdot)}(D)$ is exactly the Lebesgue space $L^p(D)$.

Define the sets $\mathcal{P}^{0}(D)$ and $\mathcal{P}(D)$ as follows $\mathcal{P}^{0}(D)=\{p(\cdot): D\rightarrow[0, \infty), p^{-}>0, p^{+}<\infty\}, $ and $\mathcal{P}(D)=\{p(\cdot): D\rightarrow[1, \infty), p^{-}>1, p^{+}<\infty\}, $where $p^{-}=ess\inf\limits_{x\in D}p(x)$ and $p^{+}=ess\sup\limits_{x\in D}p(x)<\infty$.

Throughout the paper, $p'(\cdot)$ denote the conjugate exponent of $p(\cdot), $ that is $p'(x)=p(x)/(p(x)-1).$ And note that $s'<p^{-}$ is equivalent to $s>(p')^{+}$.

Definition 1.2   Let $f\in L^{1}_{loc}(\mathbb{R}^n), $ then the Hardy-Littlewood maximal operator is defined by

$ Mf(x)=\sup\limits_{x\in B}\frac{1}{|B|}\int_{B}|f(y)|dy, $

where the supremum is taken over all balls containing $x$.

Let $\mathcal{B}(D)$ be the set of $p(\cdot)\in\mathcal{P}(D)$ such that the Hardy-Littlewood maximal operator $M$ is bounded on $L^{p(\cdot)}(D).$

Definition 1.3   [18] Let $D\subset\mathbb{R}^n$ be an open set $D\subset\mathbb{R}^n$ and $p(\cdot)\in \mathcal{P}(\mathbb{R}^n).$ $p(\cdot)$ is log-Hölder continuous, if $p(\cdot)$ satisfies the following conditions

$ |p(x)-p(y)|\leq \frac{C}{-\log(|x-y|)}, \quad x, y\in D, \quad |x-y|\leq\frac{1}{2}, \\|p(x)-p(y)|\leq \frac{C}{\log(|x|+e)}, \quad x, y\in D, \quad |y|\geq |x|, $

(1.3)

we denote by $p(\cdot)\in\mathcal{P}^{\log}(D).$ From Theorem 1.1 of [18], we know that if $p(\cdot)\in\mathcal{P}^{\log}(D), $ then $p(\cdot)\in\mathcal{B}(D).$

If $D$ is an unbounded set, we shall also use the assumption: there exists ${p(\infty)}=:\lim\limits_{|x|\rightarrow\infty}{p(x)}.$ And, we denote the subset of $\mathcal{P}^{log}(D)$ by $\mathcal{P}^{log}_{\infty}(D)$ with the exponents satisfying the following decay condition:

$ |p(x)-p(\infty)|\leq \frac{C}{\log(e+|x|)}, \quad x\in D. $

(1.4)

Note that if $D$ is an unbounded set and ${p(\infty)}$ exists, then (1.4) is equivalent to the condition (1.3). We would also like to remark that $ p(\cdot)\in \mathcal{P}^{\log}_{\infty}(D) $ if and only if $p'(\cdot)\in\mathcal{P}^{\log}_{\infty}(D) $ and $(p(\infty))' = p'(\infty), $ (see [5] for detail).

Let $D\subset\mathbb{R}^{n}$ be an open set and $\tilde{B}(x, r)=B(x, r)\bigcap D, $ where $B(x, r)$ is the ball centered at $x$ and with radius $r.$

Definition 1.4   [3] For $0<\lambda(\cdot)<n$ and $1\leq p(\cdot)<\infty, $ the Morrey space with variable exponent is defined by $L^{p(\cdot), \lambda(\cdot)}(D)=\big\{f\in L^{p(\cdot)}_{loc}(D):\|f\|_{L^{p(\cdot), \lambda(\cdot)}(D)}<\infty\big\}$, where

$ \|f\|_{L^{p(\cdot), \lambda(\cdot)}(D)}=: \sup\limits_{x\in D, r>0}r^{-\frac{\lambda(x)}{p(x)}}\|f\|_{L^p(\tilde{B}(x, r))}. $

Definition 1.5   Let $D\subset\mathbb{R}^{n}$ be an unbounded open set and $\Pi\subseteq D, \;\varphi(x, r)$ belongs to the class $\beth=\beth(\Pi\times[0, \infty))$ of non-negative functions on $\Pi\times[0, \infty)$, which are positive on $\Pi\times(0, \infty).$ Then for $1\leq p(x)\leq p^{+} < \infty, $ the generalized Morrey space with variable exponent is defined by $L^{p(\cdot), \varphi}_{\Pi}(D)=\big\{f\in L^{p(\cdot)}_{loc}(D):\|f\|_{L_{\Pi}^{p(\cdot), \varphi}}<\infty\big\}, $where

$ \|f\|_{L^{p(\cdot), \varphi}_\Pi(D)}=:\sup\limits_{x\in\Pi, r>0}\varphi(x, r)^{-\frac{1}{\theta_p(x, r)}}\|f\|_{L^{p(\cdot)}(\tilde{B}(x, r))}, $

and

$ \begin{align*} \frac{1}{\theta_p(x, r)}= \left\{ \begin{aligned} &\frac{n}{p(x), } \quad {r\leq 1, } \\ &\frac{n}{p(\infty), } \quad {r\geq 1.} \end{aligned} \right. \end{align*} $

Definition 1.6   Let $D\subset\mathbb{R}^{n}$ be an unbounded open set. The vanishing generalized Morrey space with variable exponent $VL_{\Pi}^{p(\cdot), \varphi}(D)$ is defined as the space of functions $f\in L^{p(\cdot), \varphi}_{\Pi}(D)$ such that

$ \lim\limits_{r \to 0^+}\sup\limits_{x\in\Pi}\varphi(x, r)^{-\frac{1}{\theta_p(x, r)}}\|f\|_{L^{p(\cdot)}(\tilde{B}(x, r))}=0. $

Naturally, it is suitable to impose on $\varphi(x, r)$ with the following conditions:

$ \lim\limits_{r \to 0^+}\sup\limits_{x\in\Pi}r^{\frac{n}{\theta_p{(x, r)}}}\varphi(x, r)^{-\frac{1}{\theta_p{(x, r)}}}=0 $

(1.5)

and

$ \inf\limits_{r>1}\sup\limits_{x\in\Pi}\varphi(x, r)>0. $

Noting that, if we replace $\theta_p{(x, r)}$ with $p(x)$ in Definition 1.4 and Definition 1.5, then $VL_{\Pi}^{p(\cdot), \varphi}(D)$ is the class vanishing generalized Morrey spaces with variable exponent, see [17] for example. Particularly, if $\theta_p{(x, r)}=p(x), $ $\varphi(x, r)=r^{\lambda(x)}$ and $\Pi=D$, then the generalized Morrey space with variable exponent $L^{p(\cdot), \varphi}_\Pi(D)$ is exactly the Morrey space with variable exponent $L^{p(\cdot), \lambda(\cdot)}(D).$

Definition 1.7   For $b\in L^{1}_{loc}(\mathbb{R}^n)$, then the space of functions of bounded mean oscillation is defined by $\rm{BMO}(\mathbb{R}^n)=\{f\in L^{1}_{loc}(\mathbb{R}^n):\|b\|_{BMO}<\infty\}, $where

$\|b\|_{BMO}=\sup\limits_{B}\frac{1}{|B|}\int_{B}|b(x)-b_{B}|dx$ and $b_B=\frac{1}{|B|}\int_{B}f(y)dy.$

In the following, let us state the main results of the paper.

Theorem 2.1   Assume that $D\subseteq\mathbb{R}^{n}$ is an unbounded open set. Let $0<\alpha<n, T_{\Omega, \alpha}$ be defined as in (1.1), $p(\cdot), q(\cdot)\in\mathcal{P}^{log}_{\infty}(D), $ such that $1<p^{-}\leq p^{+}< \frac{n}{\alpha}$ and $\frac{1}{q(x)}=\frac{1}{p(x)}-\frac{\alpha}{n}, x\in D.$ If $\Omega\in L^s(\mathbb{S}^{n-1})$ with $1< s'<p^{-}, $ then for arbitrary $x_0\in\Pi\subset D$ and $\tilde{B}(x_0, r)=B(x_0, r)\bigcap D, $we have

$ \|T_{\Omega, \alpha}(f)\|_{L^{q(\cdot)}(\tilde{B}(x_0, r))}\lesssim r^{\frac{n}{\theta_q{(x_0, r)}}}\int_{2r}^{\infty} t^{-\frac{n}{\theta_q{(x_0, r)}}-1}\|f\|_{L^{p(\cdot)}(\tilde{B}(x_0, t))}dt, $

(2.1)

where $s'=\frac{s}{s-1}.$

Theorem 2.2   Assume that $D\subseteq\mathbb{R}^{n}$ is an unbounded open set and $\Pi\subset D.$ Let $0<\alpha<n, T_{\Omega, \alpha}$ be defined as in (1.1), $p(\cdot), q(\cdot)\in \mathcal{P}^{log}_{\infty}(D), $ such that $1<p^{-}\leq p^{+}<\frac{n}{\alpha}$ and $\frac{1}{q(x)}=\frac{1}{p(x)}-\frac{\alpha}{n}, x\in D.$ Let $\Omega\in L^s(\mathbb{S}^{n-1})$ with $1<s'<p^{-}$ and $\varphi, \psi\in \beth=\beth(\Pi\times[0, \infty)).$ Then the operator $T_{\Omega, \alpha}$ is bounded from $VL_{\Pi}^{p(\cdot), \varphi}(D)$ to $VL_{\Pi}^{q(\cdot), \psi}(D), $ if

$ c_\delta =:\int_{\delta}^{\infty}\frac{\sup\limits_{x\in\Pi}\varphi^{\frac{1}{\theta_p{(x, t)}}}(x, t)dt}{t^{1+\frac{n}{\theta_q{(x, t)}}}}<\infty $

(2.2)

for each $\delta>0, $ and

$ \int_{r}^{\infty}\frac{\varphi^{\frac{1}{\theta_p{(x, t)}}}(x, t)dt}{t^{1+\frac{n}{\theta_q{(x, t)}}}}\leq c_0\frac{\psi^{\frac{1}{\theta_q{(x, r)}}}(x, r)}{r^{\frac{n}{\theta_q{(x, r)}}}}, $

(2.3)

where $c_0$ does not depend on $x\in \Pi$ and $r>0.$

Theorem 2.3   Assume that $D\subseteq\mathbb{R}^{n}$ is an unbounded open set. Let $0<\alpha<n, [b, T_{\Omega, \alpha}]$ be defined as in (1.2) and $p(\cdot), q(\cdot)\in \mathcal{P}^{log}_{\infty}(D), $ such that $1<p^{-}\leq p^{+}<\frac{n}{\alpha}$ and $\frac{1}{q(x)}=\frac{1}{p(x)}-\frac{\alpha}{n}, x\in D.$ If $\Omega\in L^{s}(\mathbb{S}^{n-1})$ with $1<s'<p^{-}, $ then for arbitrary $x_0\in\Pi\subset D, $ $\tilde{B}(x_0, r)=B(x_0, r)\bigcap D$ and $b\in \rm{BMO}(\mathbb{R}^n), $ we have

$ \begin{aligned} &\|[b, T_{\Omega, \alpha}](f)\|_{L^{q(\cdot)}(\tilde{B}(x_0, r))}\lesssim \|b\|_{BMO}r^{\frac{n}{\theta_q{(x_0, r)}}}\int_{2r}^{\infty}\bigg(1+\ln\frac{r}{t}\bigg)t^{-\frac{n}{\theta_q{(x_0, t)}}-1} \|f\|_{L^{p(\cdot)}(\tilde{B}(x_0, t))}dt. \end{aligned} $

(2.4)

Theorem 2.4   Assume that $D\subseteq\mathbb{R}^{n}$ is an unbounded open set and $\Pi\subset D.$ Let $0<\alpha<n, [b, T_{\Omega, \alpha}]$ be defined as (1.2), $p(\cdot), q(\cdot)\in \mathcal{P}^{log}_{\infty}(D), $ such that $1<p^{-}\leq p^{+}<\frac{n}{\alpha}$ and $\frac{1}{q(x)}=\frac{1}{p(x)}-\frac{\alpha}{n}, x\in D.$ Let $\Omega\in L^s(\mathbb{S}^{n-1})$ with $1< s'<p^{-}, b\in \rm{BMO}(\mathbb{R}^n)$ and $\varphi, \psi\in \beth=\beth(\Pi\times[0, \infty)).$ Then the commutator $[b, T_{\Omega, \alpha}]$ is bounded from $VL_{\Pi}^{p(\cdot), \varphi}(D)$ to $VL_{\Pi}^{q(\cdot), \psi}(D), $ if

$ c_\delta =:\int_{\delta}^{\infty}\bigg(1+\ln\frac{r}{t}\bigg)\frac{\sup\limits_{x\in\Pi}\varphi^{\frac{1}{\theta_p{(x, t)}}}(x, t)dt} {t^{1+\frac{n}{\theta_q{(x, t)}}}}<\infty $

(2.5)

for each $\delta>0$ and

$ \int_{r}^{\infty}\bigg(1+\ln\frac{r}{t}\bigg)\frac{\varphi^{\frac{1}{\theta_p{(x, t)}}}(x, t)dt}{t^{1+\frac{n}{\theta_q{(x, t)}}}}\leq c_0\frac{\psi^{\frac{1}{\theta_q{(x, r)}}}(x, r)}{r^{\frac{n}{\theta_q{(x, r)}}}}, $

(2.6)

where $c_0$ does not depend on $x\in \Pi$ and $r>0.$

In this part, we will give some requisite lemmas.

Lemma 3.1   (see [19, 20])(Generalized Hölder's inequality) Let $D\subseteq\mathbb{R}^n, $ $p(\cdot), q(\cdot)\in\mathcal{P}(\mathbb{R}^n), $ such that $\frac{1}{p(x)}+\frac{1}{q(x)}\equiv 1.$ If $f\in L^{p(\cdot)}(D)$ and $g\in L^{q(\cdot)}(D), $then

$ \int_{D}|f(x)g(x)|dx\leq C \|f\|_{L^{p(\cdot)}}\|g\|_{L^{q(\cdot)}}, $

with $C=\sup\limits_{x\in D}\frac{1}{p(x)} + \sup\limits_{x\in D}\frac{1}{q(x)}.$

In general, if $p_1(\cdot), p_2(\cdot)\dots p_m(\cdot)\in\mathcal{P}(\mathbb{R}^n), $ such that $\sum\limits_{k=1}^{m}\frac{1}{p_k(x)}\equiv 1, \;x\in D.$ Then for $f_i\in L^{p_i(\cdot)}(D), i=1, 2, \dots, m, $ we have

$ \int_{D}|f_1(x)\cdots f_m(x)|dx\leq C \|f_1\|_{L^{p_1(\cdot)}}\cdots\|f_m\|_{L^{p_m(\cdot)}}, $

where $C=\sum\limits_{k=1}^{m}\sup\limits_{x\in D}\frac{1}{p_k(x)}.$

Lemma 3.2   (see [21]) Let $p(\cdot), \tilde{q}(\cdot)\in \mathcal{P}(\mathbb{R}^n).$ If $p^{+}<q<\infty$ and $\frac{1}{p(x)}=\frac{1}{\tilde{q}(x)}+\frac{1}{q}, x\in\mathbb{R}^{n}, $ then we have $\|fg\|_{L^{p(\cdot)}(\mathbb{R}^n)}\lesssim\|f\|_{L^{\tilde{q}(\cdot)}(\mathbb{R}^n)}\|g\|_{L^q(\mathbb{R}^n)}, $ for all measurable functions $f$ and $g$.

Lemma 3.3   (see [5], Corollary 4.5.9) If $D$ is an unbounded set and $p\in\mathcal{P}^{log}_{\infty}(D)$, then

$ \|\chi_{B(x, r)}\|_{L^{p(\cdot)}(D)}\leq C r^{\frac{n}{\theta_{p}(x, r)}}, \quad x\in D. $

Lemma 3.4   (see [22]) Let $b\in \rm{BMO}(\mathbb{R}^n), 1<p<\infty$ and $0<r_1, r_2<\infty.$ Then

$ \bigg(\frac{1}{|B(x, r_1)|}\int_{B(x, r_1)}|b(y)-b_{B}(x, r_2)|^pdy\bigg)^{\frac{1}{p}}\leq C\bigg(1+\ln\frac{r_2}{r_1}\bigg)\|b\|_{BMO}. $

Lemma 3.5   Let $D\subseteq\mathbb{R}^n$ be an unbounded open set, $b\in \rm{BMO}(\mathbb{R}^n)$ and $p(\cdot)\in \mathcal{P}^{log}_{\infty}(D)$. Then for any $x\in D, 0<r_1, r_2<\infty$ and $\tilde{B}(x, r_i)=B(x, r_i)\bigcap D, i=1, 2, $ we have

$ \frac{1}{\|\chi_{\tilde{B}(x, r_1)}\|_{L^{p(\cdot)}(D)}}\|(b(\cdot)-b_{\tilde{B}(x, r_2)})\chi_{\tilde{B}(x, r_1)}\|_{L^{p(\cdot)}(D)} \leq C\bigg(1+\ln\frac{r_2}{r_1}\bigg)\|b\|_{BMO}. $

Proof   Let $p^{+}<p_1<\infty, p_2(x)\in\mathcal{P}^{log}_{\infty}(D), $ such that $\frac{1}{p(x)}=\frac{1}{p_1}+\frac{1}{p_2(x)}, x\in D.$ Then by Lemma 3.2, Lemma 3.3 and Lemma 3.4, we obtain

$ \begin{align*} &\frac{1}{\|\chi_{\tilde{B}(x, r_1)}\|_{L^{p(\cdot)}(D)}}\|(b(\cdot)-b_{\tilde{B}(x, r_2)})\chi_{\tilde{B}(x, r_1)}\|_{L^{p(\cdot)}(D)}\\ &\leq\frac{C}{\|\chi_{\tilde{B}(x, r_1)}\|_{L^{p(\cdot)}(D)}}\|(b(\cdot)-b_{\tilde{B}(x, r_2)})\chi_{\tilde{B}(x, r_1)}\|_{L^{p_1}(D)} \|\chi_{\tilde{B}(x, r_1)}\|_{L^{p_2(\cdot)}(D)}\\ &=C\bigg(\frac{1}{|\tilde{B}(x, r_1)|}\int_{\tilde{B}(x, r_1)}|b(y)-b_{\tilde{B}(x, r_2)}|^{p_1}dy\bigg)^{\frac{1}{p_1}} \frac{|\tilde{B}(x, r_1)|^{\frac{1}{p_1}}{\|\chi_{\tilde{B}(x, r_1)}\|}_{L^{p_2(\cdot)}(D)}}{\|\chi_{\tilde{B}(x, r_1)}\|_{L^{p(\cdot)}(D)}}\\ &\leq C\bigg(1+\ln\frac{r_2}{r_1}\bigg)\|b\|_{BMO}\frac{r_1^{\frac{n}{\theta_{p_2}{(x, r_1)}}}r_1^{\frac{n}{p_1}}}{r_1^{\frac{n}{\theta_p{(x, r_1)}}}} =C\bigg(1+\ln\frac{r_2}{r_1}\bigg)\|b\|_{BMO}. \end{align*} $

Lemma 3.6   (see [23]) Let $p(\cdot), q(\cdot)\in\mathcal{P}(\mathbb{R}^n), 0<\alpha<n, 1<p^{-}\leq p^{+}< \frac{n}{\alpha}$ and $\frac{1}{q(x)}=\frac{1}{p(x)}-\frac{\alpha}{n}$ for any $x\in\mathbb{R}^n.$ If $\frac{q(\cdot)(n-\alpha)}{n}\in\mathcal{B}(\mathbb{R}^n), \Omega\in L^s(\mathbb{S}^{n-1})$ with $1\leq s'<p^{-}, $ then there exists $C>0$ such that

$ \|T_{\Omega, \alpha}(f)\|_{L^{q(\cdot)}(\mathbb{R}^n)}\leq C\|\Omega\|_{L^s(\mathbb{S}^{n-1})}\|f\|_{L^{p(\cdot)}(\mathbb{R}^n)}. $

Lemma 3.7   (see [23]) Let $b\in \rm{BMO}(\mathbb{R}^n), p(\cdot), q(\cdot)\in\mathcal{P}(\mathbb{R}^n), 0<\alpha<n, 1<p^{-}\leq p^{+}< \frac{n}{\alpha}$ and $\frac{1}{q(x)}=\frac{1}{p(x)}-\frac{\alpha}{n}$ for any $x\in\mathbb{R}^n.$ If $\frac{q(\cdot)(n-\alpha)}{n}\in\mathcal{B}(\mathbb{R}^n), \Omega\in L^{s}(\mathbb{S}^{n-1})$ with $1<s'<p^{-}, $ then there exists a constant $C>0$ such that

$ \|[b, T_{\Omega, \alpha}](f)\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\leq C \|b\|_{BMO}\|\Omega\|_{L^s(\mathbb{S}^{n-1})}\|f\|_{L^{p(\cdot)}(\mathbb{R}^n)}. $

In this section, we will give the proofs of the main results.

Proof of Theorem 2.1   Let $f_1=f\chi_{\tilde{B}(x_0, 2r)}$ and $f_2=f\chi_{D\backslash\tilde{B}(x_0, 2r)}, $ then we have $f=f_1+f_2.$ By the sublinearity of the operator $T_{\Omega, \alpha}, $ we obtain that

$ \|T_{\Omega, \alpha}(f)\|_{L^{q(\cdot)}(\tilde{B}(x_0, r))}\leq \|T_{\Omega, \alpha}(f_1)\|_{L^{q(\cdot)}(\tilde{B}(x_0, r))} + \|T_{\Omega, \alpha}(f_2)\|_{L^{q(\cdot)}(\tilde{B}(x_0, r))}. $

(4.1)

Noting that $\frac{q(\cdot)(n-\alpha)}{n}\in\mathcal{B}(D), $ then from Lemma 3.6, we have

$ \|T_{\Omega, \alpha}(f_1)\|_{L^{q(\cdot)}(\tilde{B}(x_0, r))}\lesssim \|f_1\|_{L^{p(\cdot)}(D)}\lesssim \|f\|_{L^{p(\cdot)}(\tilde{B}(x_0, 2r))}. $

(4.2)

For the other part, let us estimate $|T_{\Omega, \alpha}(f_2)(x)|$ for $x\in\tilde{B}(x_0, r), $ first.

When $x\in \tilde{B}(x_0, r)$ and $y\in D\backslash \tilde{B}(x_0, 2r)), $ it is easy to see that $|x_0-y|\approx|x-y|.$ Let $u(x)\in \mathcal{P}^{log}_{\infty}(D), $ such that $\frac{1}{p(x)}+\frac{1}{s}+\frac{1}{u(x)}=1, x\in D.$ Then by the generalized Hölder's inequality and Lemma 3.3, we obtain

$ \begin{aligned} &|T_{\Omega, \alpha}(f_2)(x)|\leq\int_{D\backslash \tilde{B}(x_0, 2r)}\frac{|f(y)| |\Omega(x-y)|}{|x-y|^{n-\alpha}}dy \\ &\lesssim\int_{D\backslash \tilde{B}(x_0, 2r)}|f(y)| |\Omega(x-y)|\int_{|x-y|}^{\infty}t^{\alpha-n-1}dtdy\\ &\lesssim\int_{2r}^{\infty}t^{\alpha-n-1}\|f\|_{L^{p(\cdot)}(\tilde{B}(x_0, t))}\|\Omega(x-\cdot)\|_{L^{s}(\tilde{B}(x_0, t))} \|\chi_{D\bigcap\tilde{B}(x_0, t)}\|_{L^{u(\cdot)}(D)}dt\\ &\lesssim\int_{2r}^{\infty}t^{\alpha-n-1}\|f\|_{L^{p(\cdot)}(\tilde{B}(x_0, t))}\|\Omega(x-\cdot)\|_{L^{s}(\tilde{B}(x_0, t))}| \tilde{B}(x_0, t)|^{\frac{1}{\theta_u(x_0, t)}}dt. \end{aligned} $

(4.3)

When $x\in \tilde{B}(x_0, t), 2r<t<\infty, $ it's easy to see that

$ \begin{aligned} &\|\Omega(x-\cdot)\|_{L^{s}(\tilde{B}(x_0, t))}={\bigg(\int_{\tilde{B}(x-x_0, t)}|\Omega(z)|^{s}dz\bigg)}^{\frac{1}{s}}\\ &\leq {\bigg(\int_{\tilde{B}(0, t+|x-x_0|)}|\Omega(z)|^{s}dz\bigg)}^{\frac{1}{s}}\leq {\bigg(\int_{\tilde{B}(0, 2t)}|\Omega(z)|^{s}dz\bigg)}^{\frac{1}{s}}\\ &\leq {\bigg(\int_{\mathbb{S}^{n-1}}\int_{0}^{2t}|\Omega(z')|^{s}d\sigma(z')r^{n-1}dr\bigg)}^{\frac{1}{s}}\lesssim\|\Omega\|_{L^{s}(\mathbb{S}^{n-1})}|\tilde{B}(x_0, 2t)|^{\frac{1}{s}}. \end{aligned} $

(4.4)

Since $\frac{1}{q(x)}=\frac{1}{p(x)}-\frac{\alpha}{n}$ and $\frac{1}{p(x)}+\frac{1}{s}+\frac{1}{u(x)}=1, $ then for $x\in\tilde{B}(x_0, r), $ from (4.3) and (4.4), we get

$ \begin{aligned} |T_{\Omega, \alpha}(f_2)(x)| &\lesssim \int_{2r}^{\infty}t^{\alpha-n-1}\|f\|_{L^{p(\cdot)}(\tilde{B}(x_0, t))}\|\Omega\|_{L^s(\mathbb{S}^{n-1})}|\tilde{B}(x_0, 2t)|^{\frac{1}{s}} |\tilde{B}(x_0, t)|^{\frac{1}{\theta_u(x_0, t)}}dt\\ &\lesssim\int_{2r}^{\infty}t^{-\frac{n}{\theta_q{(x_0, t)}}-1}\|f\|_{L^{p(\cdot)}(\tilde{B}(x_0, t))}dt. \end{aligned} $

(4.5)

Thus,

$ \begin{aligned} \|T_{\Omega, \alpha}(f_2)\|_{L^{q(\cdot)}(\tilde{B}(x_0, r))} &\lesssim \int_{2r}^{\infty}t^{-\frac{n}{\theta_q{(x_0, t)}}-1}\|f\|_{L^{p(\cdot)}(\tilde{B}(x_0, t))}dt \|\chi_{\tilde{B}(x_0, r)}\|_{L^{q(\cdot)}(D)}\\ &\lesssim r^{\frac{n}{\theta_q{(x_0, r)}}}\int_{2r}^{\infty}t^{-\frac{n}{\theta_q{(x_0, t)}}-1}\|f\|_{L^{p(\cdot)}(\tilde{B}(x_0, t))}dt. \end{aligned} $

(4.6)

Combining the estimates of (4.1), (4.2) and (4.6) we have

$ \|T_{\Omega, \alpha}(f)\|_{L^{q(\cdot)}(\tilde{B}(x_0, r))}\lesssim r^{\frac{n}{\theta_q{(x_0, r)}}}\int_{2r}^{\infty}t^{-\frac{n}{\theta_q{(x_0, t)}}-1}\|f\|_{L^{p(\cdot)}(\tilde{B}(x_0, t))}dt, $

which completes the proof of Theorem 2.1.

Proof of Theorem 2.2   For every $f\in VL_{\Pi}^{p(\cdot), \varphi}(D)$ we need to prove that

$ \|T_{\Omega, \alpha}(f)\|_{VL_{\Pi}^{q(\cdot), \psi}(D)}\lesssim \|f\|_{VL_{\Pi}^{p(\cdot), \varphi}(D)} $

and

$ \lim\limits_{r \to 0^+}\sup\limits_{x\in \Pi}\frac{1}{\psi^{\frac{1}{\theta_q{(x, r)}}}(x, r)}\|T_{\Omega, \alpha}(f)\|_{L^{q(\cdot)}(\tilde{B}(x, r))}=0. $

In fact, from Theorem 2.1 and (2.3), it follows that

$ \begin{align*} &\|T_{\Omega, \alpha}(f)\|_{VL_{\Pi}^{q(\cdot), \psi}(D)}=\sup\limits_{x\in \Pi, r>0}\frac{1}{\psi^{\frac{1}{\theta_q{(x, r)}}}(x, r)}\|T_{\Omega, \alpha}(f)\|_{L^{q(\cdot)}(\tilde{B}(x, r))}\\ &\lesssim \sup\limits_{x\in \Pi, r>0}\frac{r^{\frac{n}{\theta_q{(x, r)}}}}{\psi^{\frac{1}{\theta_q{(x, r)}}}(x, r)}\int_{2r}^{\infty}t^{-\frac{n}{\theta_q{(x, t)}}-1}\|f\|_{L^{p(\cdot)}(\tilde{B}(x, t))}dt\\ &=\sup\limits_{x\in\Pi, r>0}\frac{r^{\frac{n}{\theta_q{(x, r)}}}}{\psi^{\frac{1}{\theta_q{(x, r)}}}(x, r)}\int_{2r}^{\infty}\frac{\varphi^{\frac{1}{\theta_p{(x, t)}}}(x, t)} {t^{1+\frac{n}{\theta_q{(x, t)}}}}\frac{\|f\|_{L^{p(\cdot)}(\tilde{B}(x, t))}}{{\varphi^{\frac{1}{\theta_p{(x, t)}}}(x, t)}}dt\\ &\lesssim \|f\|_{VL_{\Pi}^{p(\cdot), \varphi}(D)}\sup\limits_{x\in \Pi, r>0}\frac{r^{\frac{n}{\theta_q{(x, r)}}}}{\psi^{\frac{1}{\theta_q{(x, r)}}}(x, r)}\int_{r}^{\infty}\frac{\varphi^{\frac{1}{\theta_p{(x, t)}}}(x, t)} {t^{1+\frac{n}{\theta_q{(x, t)}}}}dt\lesssim \|f\|_{VL_{\Pi}^{p(\cdot), \varphi}(D)}. \end{align*} $

Now, let us show that

$ \lim\limits_{r \to 0^+}\sup\limits_{x\in \Pi}\frac{1}{\psi^{\frac{1}{\theta_q{(x, r)}}}(x, r)}\|T_{\Omega, \alpha}(f)\|_{L^{q(\cdot)}(\tilde{B}(x, r))}=0. $

For $0<r<\delta_0<\infty, $ by the estimation (2.1), it follows that

$ \frac{1}{\psi^{\frac{1}{\theta_q{(x, r)}}}(x, r)}\|T_{\Omega, \alpha}(f)\|_{L^{q(\cdot)}(\tilde{B}(x, r))}\leq C[A_{\delta_0}(x, r)+B_{\delta_0}(x, r)], $

(4.7)

where

$ \begin{align*} A_{\delta_0}(x, r):=\frac{r^{\frac{n}{\theta_q{(x, r)}}}}{{\psi^{\frac{1}{\theta_q{(x, r)}}}(x, r)}}\bigg(\int_{r}^{\delta_0} \frac{{\varphi^{\frac{1}{\theta_p{(x, t)}}}(x, t)}}{t^{1+\frac{n}{\theta_q{(x, t)}}}}\sup\limits_{0<r<t} \frac{1}{\varphi^{\frac{1}{\theta_p{(x, t)}}}(x, r)}\|f\|_{L^{p(\cdot)}(\tilde{B}(x, r))}dt\bigg), \end{align*} $

and

$ \begin{align*} B_{\delta_0}(x, r):=\frac{r^{\frac{n}{\theta_q{(x, r)}}}}{{\psi^{\frac{1}{\theta_q{(x, r)}}}(x, r)}}\bigg(\int_{\delta_0}^{\infty} \frac{{\varphi^{\frac{1}{\theta_p{(x, t)}}}(x, t)}}{t^{1+\frac{n}{\theta_q{(x, t)}}}}\sup\limits_{0<r<t} \frac{1}{\varphi^{\frac{1}{\theta_p{(x, t)}}}(x, r)}\|f\|_{L^{p(\cdot)}(\tilde{B}(x, r))}dt\bigg). \end{align*} $

For any $\varepsilon>0, $ now we choose a fixed $\delta_0$ such that whenever $0<r<\delta_0, $

$ \sup\limits_{x\in\Pi}\sup\limits_{0<r<t}\frac{1}{\varphi^{\frac{1}{\theta_p{(x, r)}}}(x, r)}\|f\|_{L^{p(\cdot)}(\tilde{B}(x, r))}<\frac{\varepsilon}{2Cc_0}, $

where $c_0$ and $C$ are constants from (2.3) and (4.7), which is possible since $f\in VL_{\Pi}^{p(\cdot), \varphi}(D).$

This allows us to estimate the first term uniformly for $0<r<\delta_0, $

$ \sup\limits_{x\in \Pi}C A_{\delta_0}(x, r)<\frac{\varepsilon}{2}. $

(4.8)

By choosing $r$ small enough, we obtain the estimate of the second term. Indeed, by (2.2), we get

$ B_{\delta_0}(x, r)\leq c_{\delta_0}\frac{r^{\frac{n}{\theta_q{(x, r)}}}}{{\psi^{\frac{1}{\theta_q{(x, r)}}}(x, r)}}\|f\|_{VL_{\Pi}^{p(\cdot), \varphi}(D)}, $

where $c_{\delta_0}$ is the constant from (2.2). Since $\psi$ satisfies the condition (1.6), we can choose $r$ small enough such that

$ \sup\limits_{x\in\Pi}\frac{r^{\frac{n}{\theta_q{(x, r)}}}}{\psi^{\frac{1}{\theta_q{(x, r)}}}(x, r)}<\frac{\varepsilon}{2Cc_{\delta_0} \|f\|_{VL_{\Pi}^{p(\cdot), \varphi}(D)}}. $

Thus,

$ \sup\limits_{x\in \Pi}C B_{\delta_0}(x, r)<\frac{\varepsilon}{2}. $

(4.9)

Combining the estimates of (4.8) and (4.9), we obtain $\frac{1}{\psi^{\frac{1}{\theta_q{(x, r)}}}(x, r)}\|T_{\Omega, \alpha}(f)\|_{L^{q(\cdot)}(\tilde{B}(x, r))}<\varepsilon.$

Therefore, $\lim\limits_{r \to 0^+}\sup\limits_{x\in \Pi}\frac{1}{\psi^{\frac{1}{\theta_q{(x, r)}}}(x, r)}\|T_{\Omega, \alpha}(f)\|_{L^{q(\cdot)}(\tilde{B}(x, r))}=0.$ The proof of Theorem 2.2 is completed.

Proof of Theorem 2.3   Let $f_1=f\chi_{\tilde{B}(x_0, 2r)}$ and $f_2=f\chi_{D\backslash\tilde{B}(x_0, 2r)}, $ then we have $f=f_1+f_2.$ Thus, it follows that

$ \|[b, T_{\Omega, \alpha}](f)\|_{L^{q(\cdot)}(\tilde{B}(x_0, r))}\leq\|[b, T_{\Omega, \alpha}](f_1)\|_{L^{q(\cdot)}(\tilde{B}(x_0, r))} +\|[b, T_{\Omega, \alpha}](f_2)\|_{L^{q(\cdot)}(\tilde{B}(x_0, r))} $

(4.10)

Noting that $\frac{q(\cdot)(n-\alpha)}{n}\in\mathcal{B}(D), $ then from Lemma 3.7, we know that the commutator $[b, T_{\Omega, \alpha}]$ is bounded in $L^{p(\cdot)}(\mathbb{R}^n), $ so

$ \|[b, T_{\Omega, \alpha}](f_1)\|_{L^{q(\cdot)}(\tilde{B}(x_0, r))}\lesssim \|f\|_{L^{p(\cdot)}(\tilde{B}(x_0, 2r))}. $

(4.11)

For the second part of (4.10), let us estimate $|[b, T_{\Omega, \alpha}](f_2)(x)|$ for $x\in\tilde{B}(x_0, r), $ first. It is easy to see that

$ \begin{align*} &|[b, T_{\Omega, \alpha}](f_2)(x)|\\ &\lesssim\int_{D\backslash\tilde{B}(x_0, 2r)}\frac{|b(x)-b_{\tilde{B}(x_0, r)}| |\Omega(x-y)| |f(y)|}{|x-y|^{n-\alpha}}dy + \int_{D\backslash\tilde{B}(x_0, 2r)}\frac{|b_{\tilde{B}(x_0, r)}-b(y)| |\Omega(x-y)| |f(y)|}{|x-y|^{n-\alpha}}dy\\ &\lesssim|b(x)-b_{\tilde{B}(x_0, r)}|\int_{D\backslash\tilde{B}(x_0, 2r)}\frac{|\Omega(x-y)| |f(y)|}{|x-y|^{n-\alpha}}dy + \int_{D\backslash\tilde{B}(x_0, 2r)}\frac{|b_{\tilde{B}(x_0, r)}-b(y)| |\Omega(x-y)| |f(y)|}{|x-y|^{n-\alpha}}dy\\ &=:I_1 + I_2. \end{align*} $

From (4.5), it is easy to see that

$ \begin{align*} &I_1\lesssim|b(x)-b_{\tilde{B}(x_0, r)}|\int_{2r}^{\infty}t^{-\frac{n}{\theta_q{(x_0, t)}}-1}\|f\|_{L^{p(\cdot)}(\tilde{B}(x_0, t))}dt. \end{align*} $

Thus, according to Lemma 3.3 and Lemma 3.5, we deduce that

$ \begin{aligned} &\|I_1\|_{L^{q(\cdot)}(\tilde{B}(x_0, r))} \\ &\lesssim \|(b(x)-b_{\tilde{B}(x_0, r)})\chi_{\tilde{B}(x_0, r)}\|_{L^{q(\cdot)}(D)}\int_{2r}^{\infty}t^{-\frac{n}{\theta_q{(x_0, t)}}-1} \|f\|_{L^{p(\cdot)}(\tilde{B}(x_0, t))}dt\\ &\lesssim \|b\|_{BMO} \|\chi_{\tilde{B}(x_0, r)}\|_{L^{q(\cdot)}(D)}\int_{2r}^{\infty}\bigg(1+\ln\frac{r}{t}\bigg)t^{-\frac{n}{\theta_q{(x_0, t)}}-1} \|f\|_{L^{p(\cdot)}(\tilde{B}(x_0, t))}dt\\ &\lesssim\|b\|_{BMO}r^{\frac{n}{\theta_q{(x_0, r)}}}\int_{2r}^{\infty}\bigg(1+\ln\frac{r}{t}\bigg) t^{-\frac{n}{\theta_q{(x_0, t)}}-1}\|f\|_{L^{p(\cdot)}(\tilde{B}(x_0, t))}dt. \end{aligned} $

(4.12)

When $x\in B(x_0, r)$ and $y\in D\backslash \tilde{B}(x_0, 2r), $ it is easy to see that $|x_0-y|\approx|x-y|.$ Let $h(x)\in\mathcal{P}^{log}_{\infty}(D), $ such that $\frac{1}{p(x)} + \frac{1}{s} + \frac{1}{h(x)} = 1, x\in D$. Then for $x \in \tilde{B}(x_0, r), $ by Lemma 3.1, we obtain

$ \begin{aligned} &I_2\lesssim \int_{D\backslash\tilde{B}(x_0, 2r)}|b_{\tilde{B}(x_0, r)}-b(y)| |\Omega(x-y)| |f(y)|\int_{|x-y|}^{\infty}\frac{dt}{t^{n-\alpha+1}}dy \\ &\quad\leq \int_{2r}^{\infty}\int_{y\in D:2r<|x-y|<t}|b_{\tilde{B}(x_0, r)}-b(y)| |\Omega(x-y)| |f(y)|dy\frac{dt}{t^{n-\alpha+1}}\\ &\quad\lesssim\int_{2r}^{\infty}|b_{\tilde{B}(x_0, r)}-b(\cdot)\|_{L^{h(\cdot)}(\tilde{B}(x_0, t))}\|\Omega(x-\cdot)\|_{L^{s}(\tilde{B}(x_0, t))} \|f\|_{L^{p(\cdot)}(\tilde{B}(x_0, t))}\frac{dt}{t^{n-\alpha+1}}. \end{aligned} $

(4.13)

From (4.4) and Lemma 3.5, we have

$ \|\Omega(x-\cdot)\|_{L^{s}(\tilde{B}(x_0, t))}\lesssim\|\Omega\|_{L^{s}(\mathbb{S}^{n-1})}|\tilde{B}(x_0, 2t)|^{\frac{1}{s}}, $

(4.14)

and

$ \|b_{\tilde{B}(x_0, r)}-b(\cdot)\|_{L^{h(\cdot)}(\tilde{B}(x_0, t))}\lesssim\bigg(1+\ln\frac{r}{t}\bigg)|\tilde{B}(x_0, t)|^{\frac{1}{\theta_h{(x_0, t)}}}. $

(4.15)

Therefore, for $x\in\tilde{B}(x_0, r)$, from (4.13), (4.14) and (4.15), we obtain

$ \begin{align*} &I_2\lesssim\|b\|_{BMO}\int_{2r}^{\infty}\bigg(1+\ln\frac{r}{t}\bigg)\|f\|_{L^{p(\cdot)}(\tilde{B}(x_0, t))} |\tilde{B}(x_0, t)|^{\frac{1}{\theta_h{(x_0, t)}}+\frac{1}{s}}t^{-n+\alpha-1}dt\\ &\quad\lesssim\|b\|_{BMO}\int_{2r}^{\infty}\bigg(1+\ln\frac{r}{t}\bigg)t^{-\frac{n}{\theta_q{(x_0, t)}}-1}\|f\|_{L^{p(\cdot)}(\tilde{B}(x_0, t))}dt. \end{align*} $

Thus,

$ \begin{aligned} &\|I_2\|_{L^{q(\cdot)}(\tilde{B}(x_0, r))} \lesssim\|b\|_{BMO}\int_{2r}^{\infty}\bigg(1+\ln\frac{r}{t}\bigg)t^{-\frac{n}{\theta_q{(x_0, t)}}-1} \|f\|_{L^{p(\cdot)}(\tilde{B}(x_0, t))}dt\|\chi_{\tilde{B}(x_0, r)}\|_{L^{q(\cdot)}(D)}\\ &\quad\lesssim\|b\|_{BMO}r^{\frac{n}{\theta_q{(x_0, r)}}} \int_{2r}^{\infty}\bigg(1+\ln\frac{r}{t}\bigg)t^{-\frac{n}{\theta_q{(x_0, t)}}-1}\|f\|_{L^{p(\cdot)}(\tilde{B}(x_0, t))}dt. \end{aligned} $

(4.16)

From (4.12) and (4.16), we get

$ \begin{aligned} \|[b, T_{\Omega, \alpha}](f_2)\|_{L^{q(\cdot)}(\tilde{B}(x_0, r))}\lesssim\|b\|_{BMO}r^{\frac{n}{\theta_q{(x_0, r)}}}\int_{2r}^{\infty}\bigg(1+\ln\frac{r}{t}\bigg)t^{-\frac{n}{\theta_q{(x_0, t)}}-1} \|f\|_{L^{p(\cdot)}(\tilde{B}(x_0, t))}dt. \end{aligned} $

(4.17)

Combining the estimates of (4.10), (4.11) and (4.17), we completed the proof of Theorem 2.3.

Proof of Theorem 2.4   The proof is similar to that of Theorem 2.2. From Theorem 2.3 and (2.6), it is easy to see that $\begin{align*} &\|[b, T_{\Omega, \alpha}](f)\|_{VL_{\Pi}^{q(\cdot), \psi}(D)}\lesssim \|b\|_{BMO}\|f\|_{VL_{\Pi}^{p(\cdot), \varphi}(D)}. \end{align*}$ So, we just have to show that

$ \lim\limits_{r \to 0^+}\sup\limits_{x\in \Pi}\frac{1}{\psi^{\frac{1}{\theta_q{(x, r)}}}(x, r)}\|[b, T_{\Omega, \alpha}](f)\|_{L^{q(\cdot)}(\tilde{B}(x, r))}=0. $

For $0<r<\delta_0<\infty.$ Then by (2.4), it follows that

$ \frac{1}{\psi^{\frac{1}{\theta_q{(x, r)}}}(x, r)}\|[b, T_{\Omega, \alpha}](f)\|_{L^{q(\cdot)}(\tilde{B}(x, r))}\leq C[I_{\delta_0}(x, r)+J_{\delta_0}(x, r)], $

(4.18)

where

$ I_{\delta_0}(x, r)=:\|b\|_{BMO}\frac{r^{\frac{n}{\theta_q{(x, r)}}}}{\psi^{\frac{1}{\theta_q{(x, r)}}}(x, r)}\int_{r}^{\delta_0} {\bigg(1+\ln\frac{r}{t}\bigg)}t^{-\frac{n}{\theta_q{(x, t)}}-1}\|f\|_{L^{p(\cdot)}(\tilde{B}(x, t))}dt, $

and

$ J_{\delta_0}(x, r)=:\|b\|_{BMO}\frac{r^{\frac{n}{\theta_q{(x, r)}}}}{\psi^{\frac{1}{\theta_q{(x, r)}}}(x, r)}\int_{\delta_0}^{\infty} {\bigg(1+\ln\frac{r}{t}\bigg)}t^{-\frac{n}{\theta_q{(x, t)}}-1}\|f\|_{L^{p(\cdot)}(\tilde{B}(x, t))}dt. $

For any $\varepsilon>0$, now we choose a fixed $\delta_0>0$ such that whenever $0<r<\delta_0, $

$ \sup\limits_{x\in \Pi}\sup\limits_{0<r<t}\frac{1}{\varphi^{\frac{1}{\theta_p{(x, r)}}}(x, r)}\|f\|_{L^{p(\cdot)}(\tilde{B}(x, r))}<\frac{\varepsilon}{2Cc_0\|b\|_{BMO}}, $

where $c_0$ and $C$ are constants from (2.6) and (4.18), respectively. It is possible since $f\in VL_{\Pi}^{p(\cdot), \varphi}(D).$ This allows us to estimate the first term uniformly for $0<r<\delta_0, $

$ \sup\limits_{x\in \Pi}CI_{\delta_0}(x, r)<\frac{\varepsilon}{2}. $

(4.19)

For the second part, from (2.5), it follows that

$ J_{\delta_0}(x, r)\leq c_{\delta_0} \|b\|_{BMO}\frac{r^{\frac{n}{\theta_q{(x, r)}}}}{\psi^{\frac{1}{\theta_q{(x, r)}}}(x, r)}\|f\|_{VL_{\Pi}^{p(\cdot), \varphi}(D)}, $

where $c_{\delta_0}$ is the constant from (2.5). Since $\psi$ satisfies the condition (1.6), it is possible to choose $r$ small enough such that

$ \sup\limits_{x\in \Pi}\frac{r^{\frac{n}{\theta_q{(x, r)}}}}{\psi^{\frac{1}{\theta_q{(x, r)}}}(x, r)}\leq\frac{\varepsilon}{2Cc_{\delta_0}\|b\|_{BMO} \|f\|_{VL_{\Pi}^{p(\cdot), \varphi}(D)}}. $

Thus,

$ \sup\limits_{x\in \Pi}C J_{\delta_0(x, r)}<\frac{\varepsilon}{2}. $

(4.20)

So, from the estimates of (4.19) and (4.20), it follows that

$ \frac{1}{\psi^{\frac{1}{\theta_q{(x, r)}}}(x, r)}\|[b, T_{\Omega, \alpha}](f)\|_{L^{q(\cdot)}(\tilde{B}(x, r))}<\varepsilon, $

which means that

$ \lim\limits_{r \to 0^+}\sup\limits_{x\in \Pi}\frac{1}{\psi^{\frac{1}{\theta_q{(x, r)}}}(x, r)}\|[b, T_{\Omega, \alpha}](f)\|_{L^{q(\cdot)}(\tilde{B}(x, r))}=0. $

Therefore, the proof of Theorem 2.4 is completed.